In triangle , and . If is a point on such that is perpendicular to , what is ?

Source: NCTM Mathematics Teacher, November 2006

**SOLUTION**

Area of triangle by Heron’s formula

Area of triangle

Area of triangle using base and altitude

**Answer**:

**Alternative solution 1**

**
**Use the Pythagorean theorem in right triangle

Similarly, in right triangle

From Eq.

Substitute the value of into Eq.

**Alternative solution 2**

**
**Use the Pythagorean theorem in right triangles and

Subtract Eq. from Eq.

Add Eq. and Eq.

Substitute the value of into Eq.

**Alternative solution 3**

Use the law of Cosine in triangle

In right triangle

Use the Pythagorean theorem in right triangle

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